For $K = \mathbb{Q}[\alpha]$ (with $\alpha$ algebraic over $\mathbb{Q}$), I understand that $\mathbb{Z}[\alpha]$ may be too coarse, and that $\mathcal{O}_K$ (the algebraic integers of $K$) allows more accurate factorizations into irreductible (not necessarily prime) factors.
But I do not understand why $\mathcal{O}_K$ is the finest subring of $K$ to be considered. Why is the definition of an algebraic integer: it has a monic minimal polynomial in $\mathbb{Z}[X]$ ? Why are no more algebraic numbers of $K$ interesting for factorizing ? Why should the set of the numbers considered be a ring (since only the multiplication matters) ?
For instance, $7 = \frac{5 + \sqrt{-3}}2 \cdot \frac{(5 - \sqrt{-3})}2$ is a "proper" factorization, while $7 = \frac{7}2 \cdot 2$ is "improper". How can a "proper" factorization be characterized ? How does this characterization define $\mathcal{O}_K$ ? I am thus looking of an alternative (and most likely equivalent) definition of $\mathcal{O}_K$, based on the idea of "proper" factorization.
Related questions: How essential is the fact that the integers of $K$ are a finitely generated $\mathbb{Z}$-module ? If $f$, a monic polynomial in $\mathbb{Z}$[X], has a monic factor in $K[X]$, does this factor already belong to $\mathcal{O}_K[X]$ ?